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A123853
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Numerators in an asymptotic expansion for the cubic recurrence sequence A123851.
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6
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1, 3, -15, 113, -5397, 84813, -3267755, 74391561, -15633072909, 465681118929, -31041303829713, 1145088996404679, -185348722911971841, 8165727090278785521, -778296382754673737187, 39898888480559205453945, -35033447016186321707305533
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OFFSET
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0,2
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COMMENTS
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A cubic analog of the asymptotic expansion A116603 of Somos's quadratic recurrence sequence A052129. Denominators are A123854.
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REFERENCES
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S. R. Finch, Mathematical Constants, Cambridge University Press, Cambridge, 2003, p. 446.
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LINKS
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T. M. Apostol, On the Lerch zeta function, Pacific J. Math. 1 (1951), 161-167. [In Eq. (3.7), p. 166, the index in the summation for the Apostol-Bernoulli numbers should start at s = 0, not at s = 1. - Petros Hadjicostas, Aug 09 2019]
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EXAMPLE
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A123851(n) ~ c^(3^n)*n^(-1/2)/(1 + 3/(4*n) - 15/(32*n^2) + 113/(128*n^3) - 5397/(2048*n^4) + ...) where c = 1.1563626843322... is the cubic recurrence constant A123852.
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MAPLE
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f:=proc(t, x) exp(sum(ln(1+m*x)/t^m, m=1..infinity)); end; for j from 0 to 29 do numer(coeff(series(f(3, x), x=0, 30), x, j)); od;
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PROG
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(PARI) {a(n) = local(A); if(n < 0, 0, A = 1 + O(x) ; for( k = 1, n, A = truncate(A) + x * O(x^k); A += x^k * polcoeff( 3/4 * (subst(1/A, x, x^2/(1-x^2))^2/(1-x^2) - 1/subst(A, x, x^2)^(2/3)), 2*k ) ); numerator( polcoeff( A, n ) ) ) } /* Michael Somos, Aug 23 2007 */
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CROSSREFS
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KEYWORD
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frac,sign
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AUTHOR
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STATUS
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approved
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